reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th58:
  (superior_setsequence(A1 (\) A)).n = (superior_setsequence A1).n \ A
proof
  reconsider X1 = (superior_setsequence(A1)) as SetSequence of X;
  reconsider X2 = (superior_setsequence(A1 (\) A)) as SetSequence of X;
A1: X1.n \ A c= X2.n
  proof
    let x be object;
    assume
A2: x in X1.n \ A;
    then
A3: not x in A by XBOOLE_0:def 5;
A4: x in X1.n by A2,XBOOLE_0:def 5;
    ex k st x in (A1 (\) A).(n+k)
    proof
      consider k such that
A5:   x in A1.(n+k) by A4,SETLIM_1:20;
      x in A1.(n+k) \ A by A3,A5,XBOOLE_0:def 5;
      then x in (A1 (\) A).(n+k) by Def8;
      hence thesis;
    end;
    hence thesis by SETLIM_1:20;
  end;
  X2.n c= X1.n \ A
  proof
    let x be object;
    assume
A6: x in X2.n;
A7: ex k st x in A1.(n+k) & not x in A
    proof
      consider k such that
A8:   x in (A1 (\) A).(n+k) by A6,SETLIM_1:20;
      x in A1.(n+k) \ A by A8,Def8;
      then x in A1.(n+k) & not x in A by XBOOLE_0:def 5;
      hence thesis;
    end;
    then x in X1.n by SETLIM_1:20;
    hence thesis by A7,XBOOLE_0:def 5;
  end;
  hence thesis by A1;
end;
